Biology Reference
In-Depth Information
Fig. 5.
Two particles of strengths
ω
1
and
ω
2
, carrying mollification kernels
ζ
.
as illustrated in Fig. 5. The core size
defines the characteristic
width of the kernel and thus the spatial resolution of the method.
The regularized function approximation is defined as
Ú
(3)
u
()
x
=
u
() (
y
z
x
-
y
)
d
y
and can be used to recover the function values at arbitrary loca-
tions
x
. The approximation error is of order
r
, hence
r
(4)
u
()
x
=
u
()
x
+
O
( ).
As introduced above,
r
is the first nonvanishing moment of the
mollification kernel.
27,85
For positive symmetric kernels, such as a
Gaussian,
r
=
2.
•
Step 3: mollified integral discretization
. The regularized integral in
Eq. (3) is discretized over
N
particles using the quadrature rule
N
(
)
Â
ω z
1
u
h
h
h
()
x
=
x
-
x
,
(5)
p
p
p
=
where
x
p
and
ω
p
are the numerical solutions of the particle positions
and strengths, determined by discretizing the ODEs in Eq. (1) in
time. The quadrature weights
ω
p
are the particle strengths and
depend on the particular quadrature rule used. The most frequent
choice is to use midpoint quadrature,
102
thus setting
ω
p
=
u
(
x
p
)
V
p
,
